latsqr(3)

NAME
latsqr - latsqr: tall-skinny QR factor

SYNOPSIS

  Functions
subroutine clatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
CLATSQR
subroutine dlatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
DLATSQR
subroutine slatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
SLATSQR
subroutine zlatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
ZLATSQR

Detailed Description

Function Documentation

  subroutine clatsqr (integer m, integer n, integer mb, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) work, integer lwork, integer info)

CLATSQR

Purpose:

    !>
    !> CLATSQR computes a blocked Tall-Skinny QR factorization of
    !> a complex M-by-N matrix A for M >= N:
    !>
    !>    A = Q * ( R ),
    !>            ( 0 )
    !>
    !> where:
    !>
    !>    Q is a M-by-M orthogonal matrix, stored on exit in an implicit
    !>    form in the elements below the diagonal of the array A and in
    !>    the elements of the array T;
    !>
    !>    R is an upper-triangular N-by-N matrix, stored on exit in
    !>    the elements on and above the diagonal of the array A.
    !>
    !>    0 is a (M-N)-by-N zero matrix, and is not stored.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrix A. M >= N >= 0.
    !>

    MB

    !>          MB is INTEGER
    !>          The row block size to be used in the blocked QR.
    !>          MB > N.
    !>

    NB

    !>          NB is INTEGER
    !>          The column block size to be used in the blocked QR.
    !>          N >= NB >= 1.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, the elements on and above the diagonal
    !>          of the array contain the N-by-N upper triangular matrix R;
    !>          the elements below the diagonal represent Q by the columns
    !>          of blocked V (see Further Details).
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    T

    !>          T is COMPLEX array,
    !>          dimension (LDT, N * Number_of_row_blocks)
    !>          where Number_of_row_blocks = CEIL((M-N)/(MB-N))
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.
    !>          See Further Details below.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T.  LDT >= NB.
    !>

    WORK

    !>          (workspace) COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
    !> representing Q as a product of other orthogonal matrices
    !>   Q = Q(1) * Q(2) * . . . * Q(k)
    !> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
    !>   Q(1) zeros out the subdiagonal entries of rows 1:MB of A
    !>   Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
    !>   Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
    !>   . . .
    !>
    !> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
    !> stored under the diagonal of rows 1:MB of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,1:N).
    !> For more information see Further Details in GEQRT.
    !>
    !> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
    !> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
    !> The last Q(k) may use fewer rows.
    !> For more information see Further Details in TPQRT.
    !>
    !> For more details of the overall algorithm, see the description of
    !> Sequential TSQR in Section 2.2 of [1].
    !>
    !> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
    !>     J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
    !>     SIAM J. Sci. Comput, vol. 34, no. 1, 2012
    !>

Definition at line 170 of file clatsqr.f.

  subroutine dlatsqr (integer m, integer n, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer lwork, integer info)

DLATSQR

Purpose:

    !>
    !> DLATSQR computes a blocked Tall-Skinny QR factorization of
    !> a real M-by-N matrix A for M >= N:
    !>
    !>    A = Q * ( R ),
    !>            ( 0 )
    !>
    !> where:
    !>
    !>    Q is a M-by-M orthogonal matrix, stored on exit in an implicit
    !>    form in the elements below the diagonal of the array A and in
    !>    the elements of the array T;
    !>
    !>    R is an upper-triangular N-by-N matrix, stored on exit in
    !>    the elements on and above the diagonal of the array A.
    !>
    !>    0 is a (M-N)-by-N zero matrix, and is not stored.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrix A. M >= N >= 0.
    !>

    MB

    !>          MB is INTEGER
    !>          The row block size to be used in the blocked QR.
    !>          MB > 0.
    !>

    NB

    !>          NB is INTEGER
    !>          The column block size to be used in the blocked QR.
    !>          N >= NB >= 1.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, the elements on and above the diagonal
    !>          of the array contain the N-by-N upper triangular matrix R;
    !>          the elements below the diagonal represent Q by the columns
    !>          of blocked V (see Further Details).
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    T

    !>          T is DOUBLE PRECISION array,
    !>          dimension (LDT, N * Number_of_row_blocks)
    !>          where Number_of_row_blocks = CEIL((M-N)/(MB-N))
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.
    !>          See Further Details below.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T.  LDT >= NB.
    !>

    WORK

    !>          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
    !> representing Q as a product of other orthogonal matrices
    !>   Q = Q(1) * Q(2) * . . . * Q(k)
    !> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
    !>   Q(1) zeros out the subdiagonal entries of rows 1:MB of A
    !>   Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
    !>   Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
    !>   . . .
    !>
    !> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
    !> stored under the diagonal of rows 1:MB of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,1:N).
    !> For more information see Further Details in GEQRT.
    !>
    !> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
    !> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
    !> The last Q(k) may use fewer rows.
    !> For more information see Further Details in TPQRT.
    !>
    !> For more details of the overall algorithm, see the description of
    !> Sequential TSQR in Section 2.2 of [1].
    !>
    !> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
    !>     J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
    !>     SIAM J. Sci. Comput, vol. 34, no. 1, 2012
    !>

Definition at line 170 of file dlatsqr.f.

  subroutine slatsqr (integer m, integer n, integer mb, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) work, integer lwork, integer info)

SLATSQR

Purpose:

    !>
    !> SLATSQR computes a blocked Tall-Skinny QR factorization of
    !> a real M-by-N matrix A for M >= N:
    !>
    !>    A = Q * ( R ),
    !>            ( 0 )
    !>
    !> where:
    !>
    !>    Q is a M-by-M orthogonal matrix, stored on exit in an implicit
    !>    form in the elements below the diagonal of the array A and in
    !>    the elements of the array T;
    !>
    !>    R is an upper-triangular N-by-N matrix, stored on exit in
    !>    the elements on and above the diagonal of the array A.
    !>
    !>    0 is a (M-N)-by-N zero matrix, and is not stored.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrix A. M >= N >= 0.
    !>

    MB

    !>          MB is INTEGER
    !>          The row block size to be used in the blocked QR.
    !>          MB > N.
    !>

    NB

    !>          NB is INTEGER
    !>          The column block size to be used in the blocked QR.
    !>          N >= NB >= 1.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, the elements on and above the diagonal
    !>          of the array contain the N-by-N upper triangular matrix R;
    !>          the elements below the diagonal represent Q by the columns
    !>          of blocked V (see Further Details).
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    T

    !>          T is REAL array,
    !>          dimension (LDT, N * Number_of_row_blocks)
    !>          where Number_of_row_blocks = CEIL((M-N)/(MB-N))
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.
    !>          See Further Details below.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T.  LDT >= NB.
    !>

    WORK

    !>          (workspace) REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
    !> representing Q as a product of other orthogonal matrices
    !>   Q = Q(1) * Q(2) * . . . * Q(k)
    !> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
    !>   Q(1) zeros out the subdiagonal entries of rows 1:MB of A
    !>   Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
    !>   Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
    !>   . . .
    !>
    !> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
    !> stored under the diagonal of rows 1:MB of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,1:N).
    !> For more information see Further Details in GEQRT.
    !>
    !> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
    !> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
    !> The last Q(k) may use fewer rows.
    !> For more information see Further Details in TPQRT.
    !>
    !> For more details of the overall algorithm, see the description of
    !> Sequential TSQR in Section 2.2 of [1].
    !>
    !> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
    !>     J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
    !>     SIAM J. Sci. Comput, vol. 34, no. 1, 2012
    !>

Definition at line 170 of file slatsqr.f.

  subroutine zlatsqr (integer m, integer n, integer mb, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) work, integer lwork, integer info)

ZLATSQR

Purpose:

    !>
    !> ZLATSQR computes a blocked Tall-Skinny QR factorization of
    !> a complex M-by-N matrix A for M >= N:
    !>
    !>    A = Q * ( R ),
    !>            ( 0 )
    !>
    !> where:
    !>
    !>    Q is a M-by-M orthogonal matrix, stored on exit in an implicit
    !>    form in the elements below the diagonal of the array A and in
    !>    the elements of the array T;
    !>
    !>    R is an upper-triangular N-by-N matrix, stored on exit in
    !>    the elements on and above the diagonal of the array A.
    !>
    !>    0 is a (M-N)-by-N zero matrix, and is not stored.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the matrix A. M >= N >= 0.
    !>

    MB

    !>          MB is INTEGER
    !>          The row block size to be used in the blocked QR.
    !>          MB > N.
    !>

    NB

    !>          NB is INTEGER
    !>          The column block size to be used in the blocked QR.
    !>          N >= NB >= 1.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, the elements on and above the diagonal
    !>          of the array contain the N-by-N upper triangular matrix R;
    !>          the elements below the diagonal represent Q by the columns
    !>          of blocked V (see Further Details).
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    T

    !>          T is COMPLEX*16 array,
    !>          dimension (LDT, N * Number_of_row_blocks)
    !>          where Number_of_row_blocks = CEIL((M-N)/(MB-N))
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.
    !>          See Further Details below.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T.  LDT >= NB.
    !>

    WORK

    !>          (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
    !> representing Q as a product of other orthogonal matrices
    !>   Q = Q(1) * Q(2) * . . . * Q(k)
    !> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
    !>   Q(1) zeros out the subdiagonal entries of rows 1:MB of A
    !>   Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
    !>   Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
    !>   . . .
    !>
    !> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
    !> stored under the diagonal of rows 1:MB of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,1:N).
    !> For more information see Further Details in GEQRT.
    !>
    !> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
    !> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
    !> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
    !> The last Q(k) may use fewer rows.
    !> For more information see Further Details in TPQRT.
    !>
    !> For more details of the overall algorithm, see the description of
    !> Sequential TSQR in Section 2.2 of [1].
    !>
    !> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
    !>     J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
    !>     SIAM J. Sci. Comput, vol. 34, no. 1, 2012
    !>

Definition at line 170 of file zlatsqr.f.

Author

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